Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $r = \dfrac{3p + 24}{p^2 - 5p} \times \dfrac{p^2 - 6p + 5}{5p + 40} $
Solution: First factor the quadratic. $r = \dfrac{3p + 24}{p^2 - 5p} \times \dfrac{(p - 5)(p - 1)}{5p + 40} $ Then factor out any other terms. $r = \dfrac{3(p + 8)}{p(p - 5)} \times \dfrac{(p - 5)(p - 1)}{5(p + 8)} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac{ 3(p + 8) \times (p - 5)(p - 1) } { p(p - 5) \times 5(p + 8) } $ $r = \dfrac{ 3(p + 8)(p - 5)(p - 1)}{ 5p(p - 5)(p + 8)} $ Notice that $(p + 8)$ and $(p - 5)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac{ 3\cancel{(p + 8)}(p - 5)(p - 1)}{ 5p\cancel{(p - 5)}(p + 8)} $ We are dividing by $p - 5$ , so $p - 5 \neq 0$ Therefore, $p \neq 5$ $r = \dfrac{ 3\cancel{(p + 8)}\cancel{(p - 5)}(p - 1)}{ 5p\cancel{(p - 5)}\cancel{(p + 8)}} $ We are dividing by $p + 8$ , so $p + 8 \neq 0$ Therefore, $p \neq -8$ $r = \dfrac{3(p - 1)}{5p} ; \space p \neq 5 ; \space p \neq -8 $